Nonparametric Identification of Risk Aversion in First-Price Auctions Under Exclusion Restrictions*

Transcription

1 Nonparametric Identification of Risk Aversion in First-Price Auctions Under Exclusion Restrictions* Emmanuel Guerre Queen Mary University of London Isabelle Perrigne Pennsylvania State University Quang Vuong Pennsylvania State University March 2007 * Financial support from the National Science Foundation grant SES is gratefully acknowledged by the last two authors. A preliminary version of this paper was presented at the 2005 CIRANO Conference on Auctions in Montreal, and the 2006 International Industrial Organization Conference in Boston. We thank participants at both conferences for their comments. Correspondence to: I. Perrigne, Department of Economics, Pennsylvania State University, University Park, PA address : perrigne@psu.edu.

2 Abstract This paper studies the nonparametric identification of the first-price auction model with risk averse bidders within the private value paradigm. We show that the benchmark model is nonindentified in general from observed bids. We derive the restrictions imposed by the model on observables and show that these restrictions are quite weak. We then establish the nonparametric identification of the bidders utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders participation leading to a latent distribution of private values independent of the number of bidders. The key idea is to exploit that the bid distribution varies with the number of bidders while the private value distribution does not. We also characterize all the theoretical restrictions imposed by such an exclusion restriction on observables to rationalize the model. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values and asymmetric bidders. Our theoretical results also extend to observed and unobserved heterogeneity. In particular, we consider endogenous bidders participation with exclusion restrictions and available instruments that do not affect the bidders private value distribution. Key words: Risk Aversion, Private Value, Nonparametric Identification, Exclusion Restrictions, Unobserved Heterogeneity JEL classification: C14, D44

3 Nonparametric Identification of Risk Aversion in First-Price Auctions Under Exclusion Restrictions E. Guerre, I. Perrigne and Q. Vuong 1 Introduction The empirical and experimental literature suggests that risk aversion is a component of bidders behavior in auctions. See Baldwin (1995), Athey and Levin (2001), Perrigne (2003) for the former and Cox, Smith and Walker (1988), Goeree, Holt and Palfrey (2002) and Bajari and Hortacsu (2005) for the latter to name a few. 1 With the exception of experimental studies, risk aversion is quite difficult to assess. The main problem lies in the identification of the bidders utility function. In a companion paper, Campo, Guerre, Perrigne and Vuong (2007) propose minimal parametric restrictions leading to the semiparametric identification of the auction model under a parameterization of the bidders utility function and a conditional quantile restriction of the private value distribution. They show that both parametric restrictions are needed. Based on this result, they derive an estimator and study its statistical properties. 2 In practice, the choice of a parametric utility function displaying risk aversion may affect the estimated results, yet various concepts of risk aversion have different implications on economic agents behavior. There is, however, no general agreement on which concept of risk aversion is the most appropriate to explain observed phenomena such as in finance through the diversification of 1 Using recent structural econometric methods, Bajari and Hortacsu (2005) find that risk aversion provides the best fit to some experimental data among a set of competing models including learning ones. 2 Since risk aversion does not affect bidding in ascending auctions within the private value paradigm, identification of risk aversion cannot be achieved in ascending auctions. The combination of first-price sealed-bid and ascending auctions, however, allows to identify nonparametrically the bidders utility function as shown by Lu and Perrigne (2006). 1

4 portfolios, insurance when low risk car drivers tend to buy more insurance than needed, or in auctions through overbidding. 3 A typical controversy is whether risk aversion is absolute or relative to economic agent s wealth. As a matter of fact, little is known on the shape of agents utility functions. Several families of utility functions have been developed to embody some economic properties related to risk aversion. See Gollier (2001) for an extensive survey on risk aversion. Which family is relevant is an empirical question. Given the importance of risk aversion in auctions and our ignorance about bidders utility functions, we address the nonparametric identification of the latter in this paper. First, we show that the general model is not identified in general from observed bids. We then derive the theoretical restrictions imposed by the model on observables and show that these restrictions are quite weak. In particular, we show that any smooth bid distribution can be rationalized by a model with general risk aversion. Such a striking result implies that risk aversion does not impose testable restrictions on bids. Second, we show that the bidders utility function is nonparametrically identified under some exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders participation leading to a latent distribution of private values that is independent of the number of bidders. Exclusion restrictions are widely used in econometrics. A typical example is the use of instrumental variables in labor economics to address the endogeneity of education in the estimation of the wage equation. Exclusion restrictions have also been used in the structural auction literature. Athey and Haile (2002) and Haile, Hong and Shum (2003) exploit some exclusion restrictions to test for common values in first-price sealed-bid auctions. Both papers assume exogenous participation to detect the winner s curse. In a different framework, Bajari and Hortacsu (2005) use exogenous participation to estimate an auction model with constant relative risk aversion from experimental data. 4 Third, we consider observed and unobserved heterogeneity. In particular, we extend our results to a model with endogenous bidders participation under exclusion restrictions and the availability of instruments that do not affect the bidders private value distribution. While considering unobserved heterogeneity affecting both bidders participation 3 For an empirical analysis of risk aversion in car insurance, see Cohen and Einav (2007). 4 Exogenous participation is not necessary to estimate the model in their paper. Such a restriction avoids the use of a conditional quantile restriction as in Campo, Guerre, Perrigne and Vuong (2007). 2

5 and the latent distribution, Haile, Hong and Shum (2003) also introduce some exogenous variables or instruments independent of the latent distribution but affecting bidders participation. Our nonparametric identification result exploits variations of the bid distribution in the number of bidders when the latter does not affect the latent private value distribution. We also characterize all the theoretical restictions on observables implied by such an exclusion restriction. In particular, we show that the rationalization of the observed bid distribution involves additional restrictions that the data must satisfy. Though we consider first a benchmark model with symmetric bidders, independent private values and no reserve price, our results extend to a binding reserve price, affiliated private values and asymmetric bidders, whether asymmetry arises from private values and/or heterogenous preferences. As such, our results apply to a large class of auction models. The paper is organized as follows. Section 2 presents the benchmark model with independent private values (IPV) and reviews the existence, uniqueness and smoothness of the equilibrium strategy. Section 3 is devoted to the identification of the benchmark model, i.e. whether its structural elements can be uniquely recovered from observed bids. We show that the model is nonidentified from observed bids. In view of this, Section 4 exploits exclusion restrictions in the form of an exogenous number of bidders to achieve nonparametric identification of the bidders utility function and private value distribution. Under such restrictions, we characterize the theoretical restrictions that observed bids need to satisfy. Section 5 extends our nonidentification and identification results to a binding reserve price, affiliated private values and asymmetric bidders. Section 6 considers observed and unobserved heterogeneity and presents a general approach for dealing with the latter. Section 7 concludes. An appendix collects the proofs. 2 The Benchmark Model This section presents the IPV first-price sealed-bid auction model with risk averse bidders and properties of its equilibrium strategy. A single and indivisible object is sold through a first-price sealed-bid auction. Within the IPV paradigm, each bidder knows his own private value v i for the auctioned object but not other bidders private values. There are I potential bidders with I Ia finite subset of {2, 3, 4,...}. Private values are drawn 3

6 independently from a distribution F ( I), which is absolutely continuous with density f( I) on a support [v(i), v(i)] IR +. The distribution F ( I) and the number of potential bidders I are assumed to be common knowledge. Let U( ) be the bidders von Neuman Morgenstern (vnm) utility function with U(0) = 0, U ( ) > 0 and U ( ) 0 thereby allowing for risk aversion. All bidders are thus identical ex ante and the game is said to be symmetric. Bidder i then maximizes his expected utility EΠ i = U(v i b i )Pr(b i b j,j i) (1) with respect to his bid b i, where b j is the jth player s bid. This corresponds to the most studied case in the auction literature where the quality of the auctioned item is known and has equivalent monetary value. See Case 1 in Maskin and Riley (1984) and Krishna (2002). 5 In addition, because the scale is irrelevant, we impose the normalization U(1) = 1. The risk neutral case is obtained when U( ) is the identity function. 6 From Maskin and Riley (1984), if a symmetric Bayesian Nash equilibrium strategy s( ) = s(, U, F, I) exists, then it is strictly increasing and continuous on [v(i), v(i)] and differentiable on (v(i), v(i)]. 7 Thus (1) becomes EΠ i = U(v i b i )F I 1 (s 1 (b i ) I), where s 1 ( ) denotes the inverse of s( ). Hence, imposing bidder i s optimal bid b i to be s(v i ) gives the following differential equation s (v i )=(I 1) f(v i I) F (v i I) λ(v i b i ) for all v i (v(i), v(i)], (2) where λ( ) =U( )/U ( ). As shown by Maskin and Riley (1984), the boundary condition is U[v(I) s(v(i))] = 0, i.e. s(v(i)) = v(i) because U(0) = 0. Moreover, the second-order conditions are satisfied. When the reserve price is nonbinding, existence of a pure equilibrium strategy follows from Maskin and Riley (2000) and Athey (2001), while its uniqueness is established by 5 Maskin and Riley (1984) consider a more general model where the utility of winning is of the form u( b i,v i ) and the utility of loosing is equal to w( ). Here, u( b i,v i )=U(v i b i ) and w(0) = U(0) = 0. 6 Bidders wealth w can be introduced in the model. The expected profit becomes [U(w + v i b i ) U(w)]Pr(b i b j,j i)+u(w). Different wealths w i lead to an asymmetric game if the w i s are common knowledge and to a multisignal game if the w i s are private information. See Che and Gale (1998) for a multisignal auction model with budget constraints. 7 Moreover, as noted by Maskin and Riley (1984, Remark 2.3), the only equilibria are symmetric when F ( ) has bounded support, which is assumed below. 4

7 Maskin and Riley (2003) using an argument similar to Lebrun (1999). The main contribution of Theorem 1 below is to derive the smoothness of the equilibrium strategy, which is used in the next section. Determining the smoothness of the equilibrium strategy is difficult when the differential equation (2) does not have an explicit solution, which is the case for general utility functions U( ). This is more so as (2) is known to have a singularity at v(i) when the reserve price is nonbinding. To address these difficulties we rewrite (2) as a differential equation in the bid quantile function b(α, I) =s[v(α, I)], where α [0, 1] and v(α, I) is the α-quantile of F ( I). We then view the latter differential equation as a member of a set (also called flow) of differential equations E(B; t) = 0 parameterized by t [0, 1] in an unknown function B( ), where E(B; 1) = 0 corresponds to the general utility function U( ), while E(B; 0) = 0 corresponds to an appropriate constant relative risk aversion (CRRA) utility function. See (B.1) (B.3) in Appendix B. Next, we adopt a functional approach which exploits the existence, uniqueness, and smoothness of the equilibrium strategy in the CRRA case, where the solution of (2) is known explicitly. In particular, our functional approach delivers the existence and uniqueness of the equilibrium strategy for a general utility function U( ) by a Continuation Argument Theorem, thereby providing an alternative proof to those used in the economics literature. Moreover, our framework establishes the smoothness of the equilibrium strategy by an Implicit Functional Theorem. We assume that U( ) and F ( I) belong to U R and F R defined as follows, respectively. Definition 1: For R 1, let U R be the set of utility functions U( ) satisfying (i) U :[0, + ) [0, + ), U(0) = 0 and U(1) = 1, (ii) U( ) is continuous on [0, + ), and admits R +2continuous derivatives on (0, + ) with U ( ) > 0 and U ( ) 0 on (0, + ), (iii) lim x 0 λ (r) (x) is finite for 1 r R +1, where λ (r) ( ) is the rth derivative of λ( ). Conditions (i) and (ii) have been discussed previously. Note that lim x 0 λ(x) = 0 since U(0) = 0 and U ( ) is nonincreasing. Thus, from (ii) and (iii) it follows that λ( ) admits R + 1 continuous derivatives on [0, + ). These regularity assumptions are weak as they are satisfied by many vnm utility functions. Definition 2: For R 1, let F R be the set of distributions F ( I), I I, satisfying (i) F ( I) is a c.d.f. with support of the form [v(i), v(i)], where 0 v(i) < v(i) < +, 5

8 (ii) F ( I) admits R +1continuous derivatives on [v(i), v(i)], (iii) f( I) > 0 on [v(i), v(i)]. Altogether (i) (iii) imply that f( I) is bounded away from zero on [v(i), v(i)]. Theorem 1: Let R 1. Suppose that [U, F ] U R F R, then for each I I, there exists a unique (symmetric) equilibrium strategy s( ). Moreover, this strategy satisfies: (i) v (v(i), v(i)], s(v) <vwith s(v(i)) = v(i), (ii) v [v(i), v(i)], s (v) > 0 with s (v) =(I 1)λ (0)/[(I 1)λ (0) + 1] < 1, (iii) s( ) admits R +1 continuous derivatives on [v(i), v(i)]. The proof of Theorem 1 can be found in Appendix B. 3 General Nonidentification Results In this section we study identification of the structure [U, F ] from observables. We assume that the number I of bidders is observed, as in a first-price sealed-bid auction with a nonbinding reserve price. We also assume that the distribution G( I) of an equilibrium bid is known. Thus the identification problem reduces to whether the structure [U, F ] can be recovered uniquely from the knowledge of (I,G). A related issue is whether any bid distribution G( I) can be rationalized by a structure [U, F ]. Such a question relates to the possibility of testing the validity of the auction model under consideration. Following Guerre, Perrigne and Vuong (2000), we express (2) using the distribution G( I) of an equilibrium bid. For every b [b(i), b(i)] = [v(i),s(v(i))], we have G(b I) = F (s 1 (b) I) =F (v I) with density g(b I) =f(v I)/s (v), where v = s 1 (b). Thus (2) can be written as 1=(I 1) g(b i I) G(b i I) λ(v i b i ) for all b i (b(i), b(i)]. (3) Since U( ) 0 and U ( ) 0, we have λ ( ) =1 U( )U ( )/U 2 ( ) 1. Thus λ( ) is strictly increasing. Solving (3) for v i and using b(i) =v(i) with λ 1 (0) = 0 give ( ) 1 v i = b i + λ 1 G(b i I) ξ(b i,u,g,i) for all b i [b(i), b(i)], (4) I 1 g(b i I) 6

9 where λ 1 ( ) denotes the inverse of λ( ). This equation expresses each bidder s private value as a function of his corresponding bid, the bid distribution, the number of bidders and the utility function. Note that ξ( ) is the inverse of the bidding strategy s( ). The equilibrium bid distribution G( I) satisfies some regularity properties implied by the smoothness of s( ) given in Theorem 1 and the regularity assumptions on [U, F ]. Definition 3: For R 1, let G R be the set of distributions G( I), I I, satisfying (i) G( I) is a c.d.f. with support of the form [b(i), b(i)], where 0 b(i) < b(i) < +, (ii) G( I) admits R +1 continuous derivatives on [b(i), b(i)], (iii) g( I) > 0 on [b(i), b(i)], (iv) g( I) admits R +1 continuous derivatives on (b(i), b(i)], (v) lim b b(i) d r [G(b I)/g(b I)]/db r exists and is finite for r =1,...,R+1. The regularity properties (i) (iii) are similar to those of Definition 2 for F ( I). They imply that g( I) is bounded away from zero on [b(i), b(i)] and lim b b G(b I)/g(b I) = 0 so that lim b b(i) ξ(b, U, G, I) =b(i). Properties (iv) and (v) are specific to the auction model. In particular, (iv) says that g( I) is smoother than f( I), extending a similar property noted by Guerre, Perrigne and Vuong (2000) for the risk neutral model. Combined with (iii) and (iv), (v) implies that G( I)/g( I) isr + 1 continuously differentiable on [b(i), b(i)]. The following lemma provides necessary and sufficient conditions for rationalizing a distribution of observed bids by an IPV auction model with risk aversion. Hereafter, we say that a distribution is rationalized by an auction model with risk aversion if there exists a structure [U, F ] whose equilibrium bid distribution is identical to the given distribution. Lemma 1: Let R 1, and G( I) be the joint distribution of (b 1,...,b I ) conditional on I I. There exists an IPV auction model with risk aversion [U, F ] U R F R that rationalizes G( ) if and only if (i) G(b 1,...,b I I) = I i=1 G(b i I), with G( ) G R, (ii) λ : IR + IR + with R +1 continuous derivatives on [0, + ), λ(0) = 0 and λ ( ) 1 such that ξ ( ) > 0 on [b(i), b(i)], where ξ(b, U, G, I) =b + λ 1 [G(b I)/((I 1)g(b I))]. Condition (i) is related to the IPV paradigm and requires that bids be i.i.d., where G( ) satisfies the regularity properties of Definition 3. Condition (ii) arises from ξ(,u,g,i) being the inverse of the equilibrium strategy, which is strictly increasing for each I I. 8 8 As shown in the proof of Lemma 1, if condition (ii) is satisfied, then G( I) is rationalized by the 7

10 The next proposition shows that any distribution G( ) G R can be rationalized by an IPV auction model with a utility function displaying risk aversion. Proposition 1: Let R 1. A bid distribution G( ) can be rationalized by a risk averse structure [U, F ] U R F R if and only if G( ) G R. Proposition 1 is striking. It implies that the restriction (ii) in Lemma 1 for rationalizing a bid distribution with risk averse bidders is redundant. Specifically, our proof indicates that we can always find a function λ( ) corresponding to a utility function U( ) U R so that condition (ii) in Lemma 1 is satisfied whenever G( ) G R. Alternatively, the IPV auction model with general risk aversion does not impose any restriction on observed bids beyond their independence and the weak regularity conditions embodied in G R. Thus, by allowing for risk aversion, one does enlarge the set of rationalizable bid distributions relative to the risk neutral case studied in Guerre, Perrigne and Vuong (2000). 9 A model is a set of structures [U, F ]. Hereafter, a structure [U, F ]isnonidentified if there exists another structure [Ũ, F ] within the model that leads to the same equilibrium bid distribution. If no such structure [Ũ, F ] exists for any [U, F ], the model is (globally) identified. Given the weakness of the restrictions imposed by the model, it is not surprising that the model with general risk aversion is not identified. Proposition 2: Let R 1. Any structure [U, F ] U R F R is not identified. Proposition 2 implies that the auction model with risk averse bidders is nonparametrically nonidentified. This contrasts with Guerre, Perrigne and Vuong (2000) who show that the auction model with risk neutral bidders is nonparametrically identified. Thus the nonidentification of the general risk aversion model U R F R arises from the unknown utility function U( ), which is restricted to the identity function under risk neutrality. In view of this result, one can entertain several strategies to identify the model. A first natural strategy is to require more data. For instance, the availability of ascending structure [U, F ], where U(x) = exp x (1/λ(t))dt and F ( I) is the distribution of ξ(b, U, G, I) with b 1 G( I). Because λ(x) λ (0)x in the neighborhood of 0, 0 1 (1/λ(t))dt = so that U(0) = 0, as required. 9 Campo, Guerre, Perrigne and Vuong (2007) show another interesting result: Any distribution G( ) G R can be rationalized by some constant relative or absolute risk aversion model. Thus, allowing for constant relative or absolute risk aversion can explain any smooth bid distribution. 8

11 auction data allows to identifiy nonparametrically [U, F ] as shown in Lu and Perrigne (2006). A second strategy is to impose more restrictions on the structure [U, F ] through a parameterization of U( ) and/or F ( ). This is pursued in Campo, Guerre, Perrigne and Vuong (2007) where U( ) and one quantile of F ( ) are parameterized, which are minimal parametric assumptions to identify semiparametrically the model. Hereafter, we exploit another type of restrictions, namely exclusion restrictions, which have been used extensively in econometrics for identifying demand and supply as well as sample selection models in labor economics. 4 Nonparametric Identification and Restrictions We now assume that F ( ) does not depend on the number I of bidders, which corresponds to the restriction F ( I) = F ( ). As such, bidders participation is exogenous. The functions U( ) and F ( ) satisfy the regularity conditions of Definitions 1 and 2 with R 1. The previous definitions and equations defining the model need to be revised accordingly with F ( ) and f( ) defined on support [v,v], while the bid distribution G( I) still depends on I through s(,u,f,i) but its support is now [v,s(v)], where the upper bound depends implicitly on I. The key idea of our nonparametric identification result is to exploit variations in the quantiles of the bid distribution with the number of bidders, while the corresponding quantiles of the private value distribution remain the same. Our result relies on a property of the equilibrium strategy, namely that s( ) is increasing in bidders participation. In simple terms, increased competition renders bidding more aggressive. 10 Let I 2 >I 1 2 be two different numbers of bidders. We use the index j =1, 2 to refer to the level of competition. Because the equilibrium strategy defined in (2) varies with the number of bidders, the bid distribution will also vary with the number of bidders giving s j ( ) and G j ( ) G( I j ). Though the lower bound of the bid distribution remains the same because of the boundary condition, the upper bound b j varies with competition. The next lemma gives some lower and upper bounds for each equilibrium strategy in terms of the other equilibrium strategy. In particular, it establishes that the equilibrium strategy strictly increases in the number of bidders. As far as we know, the latter property was 10 Identification of the bidders utility function when the equilibrium strategies are nonincreasing in competition is discussed in Section

12 obtained for the risk neutral case and the CRRA case, but not when risk aversion takes the general form U( ). 11 Lemma 2: Under the previous assumptions, we have for any v (v, v]. I 1 1 I 2 1 s 2(v)+ I 2 I 1 I 2 1 v <s 1(v) <s 2 (v) < I 2 1 I 1 1 s 1(v)+ I 1 I 2 I 1 1 v The preceding lemma provides some testable implications in terms of stochastic dominance between the observed equilibrium bid distributions as well as their quantiles. Let G 1 ( ) b G 2 ( ) denote that the distribution G 1 ( ) is strictly (first-order) stochastically dominated by G 2 ( ) except at the common lower bound b of their supports. That is, G 1 (b) >G 2 (b) for any b (b, b 1 ], where the support of G j ( ) is[b, b j ], which is a compact subset with nonempty interior of [0, ). For j =1, 2, let b j (α) denote the α-quantile of the equilibrium bid distribution G j ( ), i.e. G j [b j (α)] = α for α [0, 1]. Because b j = s j (v) where s j ( ) is strictly increasing on [v, v], b j (α) =s j [v(α)], where v(α) is the α-quantile of F ( ). Hence, from Lemma 2 the quantiles of G 1 ( ) and G 2 ( ) satisfy I 1 1 I 2 1 b 2(α)+ I 2 I 1 I 2 1 b <b 1(α) <b 2 (α) < I 2 1 I 1 1 b 1(α)+ I 1 I 2 I 1 1 b (5) for any α [0, 1]. Equivalently, let G jk ( ) denote the distribution of [(I k 1)b j +(I j I k )b]/[i j 1], where j, k =1, 2, and b j = s j (v). 12 Thus, the lower bound of the support of G jk ( ) isb and we have G 21 ( ) b G 1 ( ) b G 2 ( ) b G 12 ( ). When the number I of bidders can take more than two values, the previous results imply several testable stochastic dominance relations among the observed bid distributions associated with the different values for I. Several of them are actually redundant. For instance, suppose that I [I, I] with 2 I < I<. The above implies that there are 4[ (I I)] = 2(I I)(I I + 1) stochastic dominance relations. The next corollary shows that there are at most 2(I I + 1) relevant relations Only the middle inequality will be used for establishing the nonparametric identification of [U( ),F( )]. 12 When j = k, G jk = G j ( ). 13 See Barrett and Donald (2003) for consistent tests of stochastic dominance. 10

13 Corollary 1: Suppose that I I [I, I] with 2 I < I. Under the previous assumptions, the quantiles b(α, I) of the equilibrium bid distribution G( I) satisfy { max b(α, I 1), I 1 } { I b(α, I+1)+1 I b <b(α, I)<min b(α, I+1), I 1 } 1 b(α, I 1) I 2 I 2 b for any α (0, 1] and any I [I, I]. 14 Equivalently, let b(i) denote the equilibrium bid with I bidders. Let G I ( ) denote the distribution of the maximum of b(i 1) and [(I 1)b(I +1)+b]/I and G I ( ) denote the distribution of the minimum of b(i +1)and [(I 1)b(I 1) b]/(i 2). Hence, G I ( ) b G( I) b G I ( ), for any I [I, I]. Given two different numbers of bidders I 2 >I 1, we now turn to the nonparametric identification of [U( ),F( )] or equivalently [λ( ),F( )] as U(x) = exp x 1 1/λ(t)dt using the normalization U(1) = 1. Specifically, our proof is constructive and shows that the inverse function λ 1 ( ), which exists because λ( ) is strictly increasing on [0, + ), is nonparametrically identified on the range R 1 of the function R 1 (α), where α [0, 1] and R j (α) = 1 α (6) I j 1 g j [b j (α)] for j =1, 2. Note that the range R j of R j ( ) is of the form [0, r j ] with 0 < r j < because g j ( ) is bounded away from zero and continuous on [0, b j ] by Definition 3 and Lemma 1. Moreover, note that R j (α) =λ[v j (α) s j (v(α))] from (4). Thus, identifying nonparametrically λ 1 ( ) onr j is equivalent to identifying nonparametrically λ( ) on the range of the markdown/rent v s j (v), where v [v, v]. Because s 1 ( ) <s 2 ( ) on(v, v] by Lemma 2, we have R 1 ( ) >R 2 ( ) on(0, 1]. Thus, r 2 < r 1 so that R 2 is strictly included in R 1. Hence, the risk aversion function λ( ) is identified nonparametrically on the largest set of possible markdowns [0, max v [v,v] v s 1 (v)]. 15 The next proposition provides explicit expressions for λ( ) and F ( ). Proposition 3: Under the previous assumptions, λ 1 ( ) is identified nonparametrically on R 1. Specifically, λ 1 (0) = 0 and for any u 0 R 1 \{0}, λ 1 ( ) is given by λ 1 (u 0 )= + t=0 b(α t ), 14 Obviously, b(,i 1) is dropped when I = I, while b(,i + 1) is dropped when I = I. 15 In general, max v [v,v] v s j (v) v s j (v). On the other hand, if the markdown v s j (v) is increasing in v, then max v [v,v] v s j (v) =v s j (v). Moreover, R j ( ) would be increasing in α and r j = R j (1). 11

14 where b(α) = b 2 (α) b 1 (α), and the sequence {α t } is strictly decreasing with 0 < α t 1 satisfying the nonlinear recursive relation R 1 (α t )=R 2 (α t 1 ) with initial condition R 1 (α 0 )=u 0. Moreover, F ( ) is identified nonparametrically on [v, v] with F ( ) = G j [ξj 1 ( )] for j =1, 2. The sequence {α t } is not necessarly unique. Proposition 3 explains how to construct such a sequence recursively. The key idea is to use the invariance of the quantile of the private value distribution v(α) for two different numbers of bidders I 1 and I 2. Specifically, using (4) leads to the compatibility condition λ 1 [R 1 (α)] = λ 1 [R 2 (α)] + b(α), where b(α) > 0 by Lemma 2. Because R 1 (0) = 0 and R 1 ( ) >R 2 ( ) on (0, 1], the continuity of R 1 ( ) implies that there exists a value α such that α <αand R 1 ( α) =R 2 (α), which can be used to rewrite the preceding compatibility condition. But the latter also holds at α. Continuing the same exercise gives the sequence of values α t. We show that there exists at least one such sequence {α t }. When R 1 ( ) is strictly increasing, i.e. when the markdown or bidders rent with I 1 bidders is strictly increasing in v, such a sequence is unique. When R 1 ( ) is not strictly increasing, the sequence {α t } may not be unique, but all such sequences must lead to the same value for t=0 b(α t ), which then defines λ 1 (u 0 ) uniquely. The construction of such a sequence is illustrated in Figure 1. Figure 1 displays the equilibrium strategies s 1 ( ) <s 2 ( ). For α 0 (0, 1], consider the α 0 -quantile v(α 0 )off( ). The markdown v(α 0 ) b 1 (α 0 ) is the sum of b(α 0 ), which is known and λ 1 [R 2 (α 0 )], which is unknown. The latter is equal to the markdown v(α 1 ) b 1 (α 1 ), which is also the sum of b(α 1 ) and λ 1 [R 2 (α 1 )]. Continuing this construction gives the sequence {α t } and establishes the unknown component λ 1 [R 2 (α 0 )] as the infinite series of known differences in bid quantiles (α t ). An important related question is to characterize all the restrictions on the observed equilibrium bid distributions that arise from the independence of the private value distribution F ( ) on the number I of bidders. In particular, it is useful to assess whether the observed bid distributions, which typically vary with the number I of bidders, can be rationalized by a structure [U( ),F( )] that is independent of I. In other words, these restrictions allow to test the validity of the model and its assumptions. Violation of one of these restrictions leads to reject the model for explaining the observed bids. In partic- 12

15 ular, it could mean that the exogeneity of bidders participation is not justified. Lemma 3 provides such restrictions when I takes two different values I 2 >I 1. Lemma 3: Let I = {I 1,I 2 } with I 1 <I 2. Let G j (,..., ) be the joint distribution of (b 1,...,b Ij ), j =1, 2. The equilibrium bid distributions G j ( ), j =1, 2, are rationalized by a structure [U( ),F( )] independent of I if and only if (i) For each j =1, 2, G j (b 1,...,b Ij )= I j i=1 G j (b i ), where G j ( ) =G( I j ) with support of the form [b, b j ] and {G( I); I I} G R, (ii) The α-quantiles of G 1 ( ) and G 2 ( ) satisfy b 1 (α) <b 2 (α) for α (0, 1], i.e. G 1 ( ) b G 2 ( ), (iii) λ( ) :IR + IR + with R +1 continuous derivatives on [0, + ), λ(0) = 0 and λ ( ) 1 such that (a) the compatibility condition is satisfied for any α [0, 1], namely, ( ) ( ) 1 b 2 (α)+λ 1 α 1 = b 1 (α)+λ 1 α, (7) I 2 1 g 2 (b 2 (α)) I 1 1 g 1 (b 1 (α)) (b) for I j I, ξ j ( ) > 0 on [b, b j], where ξ j (b) =b + λ 1 [G j (b)/((i j 1)g j (b))]. Unlike Lemma 2, which only provides some (testable) implications, Lemma 3 characterizes all the theoretical restrictions imposed by the model with an exogenous bidders participation. Relative to the general case of Section 3 in which F ( ) can vary with I, the set of bid distributions that can be rationalized is much reduced because of the restrictions (ii) and (iii)(a). Indeed, Lemma 1 implies that any distribution G j ( ) G R can be rationalized by a structure [U( ),F( I j )], which is not identified. Thus, these additional restrictions help in identifying nonparametrically the structure [U, F ]. 16 As in Corollary 1, Lemma 3 can be extended straightforwardly to the case where I I [I, I]. Specifically, (i) and (iii)-(b) hold, while (ii) and (iii)-(a) still hold for all pairs (I j,i k ) I,k j. 5 Extensions This section extends our results to a binding reserve price, affiliated private values and asymmetric biddders in private values and/or preferences. Except for the first part of 16 The above compatibility conditions are similar in spirit to the ones used to identify the model with risk aversion and heterogenous preferences. See Campo, Guerre, Perrigne and Vuong (2007). 13

16 Proposition 7, we do not provide formal proofs of Propositions 4 7 though we indicate how they can be established in the text. 5.1 Binding Reserve Price A binding reserve price, i.e. p 0 >v, introduces a truncation in the observed bid distribution as only the I bidders who have a value above p 0 will bid at the auction. Let G ( I) be the truncated bid distribution on [p 0, b(i)]. We observe I the number of active bidders, I I,I I. Because G (b I) =[F (v I) F (p 0 I)]/[1 F (p 0 I)] for b [p 0, b(i)], elementary algebra gives the following inverse equilibrium strategy ( 1 v = s 1 (b )=b +λ 1 G (b I) I 1 gj (b I) + 1 I 1 ) 1 F (p 0 I) g (b I) 1 F (p 0 I) ξ(b,u,g,i,f(p 0 I)). (8) Definitions 1, 2 and 3 remain the same with the exception that p 0 replaces b(i) in Definition 3. Moreover, because lim b p0 g (b I) = + as s (p 0 ) = 0 from (2), we allow derivatives and limits to be infinite at p 0 in Definition Given that I is Binomial distributed with parameters [I,1 F(p 0 I)], I and F (p 0 I) are identified. This identification applies on subsets of auctions. Proposition 4: Any structure [U, F ] U R F R with a binding reserve price is not identified. On the other hand, the structure [U, F ] with the exclusion restriction F ( I) = F ( ) is identified. Namely, U( ) is identified on [0, max v [v,v] v s 1 (v)], while F ( ) is identified on [p 0, v]. The observed bid distribution G (,..., ) is rationalized if only if Lemma 1 is satisfied with G R and ξ( ) as defined above. From this rationalization result, any G ( I) G R,I I can be rationalized by a risk averse structure [U, F ] U R F R. It is then straightforward to show that the structure [U, F ] U R F R is not identified. Under the exogeneity of the number of bidders, we assume that we identify at least two levels of potential bidders I 1 <I 2. Let G j ( ) be the truncated bid distribution on 17 To avoid infinite derivatives/limits at p 0, we can consider the bid transformation used in Guerre, Perrigne and Vuong (2000, Section 4), in which case rationalization and identification are based on the density of the transformed bids. 14

17 [p 0, b j ]. Note that Lemma 2 still holds with a binding reserve price as the latter simply reduces the shading relative to the case with no reserve price. In view of (8), the function R j (α) becomes R j (α) = 1 ( 1 α + F (p ) 0), I j 1 gj [b j(α)] 1 F (p 0 ) for j =1, 2. Note that R j (α) differs from (6) by the additional term F (p 0 )/(1 F (p 0 )). As before, the number of potential bidders I j and F (p 0 ) are identified from the distribution of the number of actual bidders. The problem reduces to identifying λ 1 ( ) and F ( ) on [0, r 1 ] and [p 0, v], respectively. A simple extension of Proposition 3 shows that [λ 1 ( ),F( )] is nonparametrically identified on these intervals using the quantiles b j (α) of G j ( ). Similarly, Lemma 3 can be readily adapted. 5.2 Affiliated Private Values The vector (v 1,...,v I ) is distributed as F (,..., I), which is exchangeable in its I arguments, affiliated and defined on [v(i), v(i)] I. We follow the notations of Li, Perrigne and Vuong (2002). Let G Bi b i (b i b i,i) be the probability that i has a bid larger than all his opponents conditional on his bid b i with B i = max k i b k and b i = s(v i ). Without loss of generality, we can consider G B1 b 1 (,I) as all bidders are symmetric. To simplify the notations, we omit the index 1. The inverse equilibrium strategy becomes ( ) v = s 1 (b) =b + λ 1 GB b (b b, I) ξ(b, U, G,I) for all b [b(i), b(i)] (9) g B b (b b, I) with the joint bid distribution G(,..., I). Definitions 1 and 2 remain the same except that F (,..., I) isr + I continuously differentiable following Li, Perrigne and Vuong (2000, 2002). Note that G B b ( I)/g B b ( I) =G B b (, I)/g Bb (, I), where G B b (, I) G Bb (, I)/ b and g Bb (, I) are the b-derivative of the joint c.d.f. and the joint density of (B,b), respectively. Let G R be the set of exchangeable and affiliated distributions {G(,..., I),I I}with R continuously differentiable densities such that G B b (b, b I)/g Bb (b, b I) isr + 1 continuously differentiable in b [b(i), b(i)] and strictly positive on (b(i), b(i)]. Proposition 5:Any structure [U, F ] U R F R with affiliated values is not identified. On the other hand, the structure [U, F ] with the exclusion restriction F (,..., I) =F (,..., ) 15

18 is identified. Namely, U( ) is identified on [0, max v [v,v] v s 1 (v)], while F (,..., ) is identified on [v, v] I. The observed bid distribution G(,..., ) is rationalized if and only Lemma 1 is satisfied with G R and ξ( ) as defined above. From this rationalization result, any G(,..., ) G R can be rationalized by some risk averse structure [U, F ] U R F R. We can then show that the structure [U, F ] U R F R is not identified using a similar argument as in Proposition 2, where G( I)/[(I 1)g( I)] is replaced by G B b (, I)/g Bb (, I) in view of (4) and (9). Under two competitive environments I 1 <I 2 and exogenous bidders participation, the vector (v 1,...,v Ij ) is distributed as F j (,..., ), which is exchangeable in its I j arguments and affiliated. Moreover, as participation is exogenous, F 1 (,..., ) and F 2 (,..., ) are related by F 1 (v 1,...,v I1 )= v v... v v F 2(v 1,...,v I1,v I1 +1,...,v I2 )dv I dv I2, i.e. F 1 (,..., ) is the marginal of F 2 (,..., ). Hence, F j (,..., ) has support [v, v] j. Assume that the structures [U, F j ],j =1, 2 satisfy s 1 (v) <s 2 (v), i.e. competition renders bidding more aggressive. 18 Here again, the exogeneity of the number of bidders allows us to identify λ 1 ( ) on [0, r 1 ] by exploiting variations in the number of bidders, where R j ( ) =G j B b (b j(α),b j (α))/g j B,b (b j(α),b j (α)) with b j (α) the α-quantile of the marginal bid density g j ( ) associated with I j bidders. Specifically, Proposition 3 and Lemma 3 similarly extend to this case. 5.3 Asymmetric Bidders Asymmetry among bidders, which is known ex ante to all participants, can arise from two different sources, namely from (i) different distributions of private values and/or (ii) different utility functions. We consider these cases separately. Asymmetry in Private Values Given I I, the joint private value distribution is F(,..., I) = i F i ( I) with each F i ( I) satisfying Definition 2 on the support [v(i), v(i)]. To simplify, we assume that all F i ( I) have the same support. Let F R be the set of such distributions F(,..., I) when I I. Because of the boundary conditions s i (v(i)) = v(i) and s i (v(i)) = s j (v(i)), 18 Section 5.4 relaxes this assumption. The competition effect is unclear with affiliated private values as some distributions F (,..., ) may lead to bidding strategies decreasing in the number of bidders as shown by Pinkse and Tan (2005). 16

19 j i, bidder i s distribution G i ( I) is defined on [b(i), b(i)] for all i =1,...,I. Following Campo, Perrigne and Vuong (2003), we have ( ) v i = b i + λ 1 1 ξ i (b i,u,g,i), where H i ( I) = g j ( I) H i (b i I) j i G j ( I), (10) for i =1,...,I. Let G R be the set of distributions G(,..., ) such that each marginal distribution G i ( ) satisfies Definition 3 with G(b I)/g(b I) replaced by 1/H i (b I) in (v). Proposition 6: Any structure [U, F] U R F R with asymmetry in private values is not identified. On the other hand, the structure [U, F] with the exclusion restriction F i ( I) = F i ( ), i Iis partially identified. Namely, U( ) is identified on [0, max v [v,v],i=1,...,i0 v s i (v)], while F i ( ),i =1,...,I 0 are identified on [v, v], where I 0 is the number of bidders participating to both auctions. For the remaining bidders, F i ( ) is identified for the quantiles satisfying R i (α) [0, max j=1,...,i0 r j1 ], where R i (α) is defined below. The bid distribution G(,..., ) is rationalized if and only if Lemma 1 is satisfied with G R and ξ i ( ),i=1,...,i as defined above. Hence, any G(,..., ) G R can be rationalized by a structure with [U, F] U R F R. It is then straigthforward to show that any structure [U, F] U R F R is not identified. Under two competitive environments I 2 >I 1 2 and exogenous bidders participation, the bidder of type i has the same private value distribution irrespective of the number of bidders participating to the auction. Thus, all F i ( I)s are defined on the same support [v, v]. Since our results under exclusion restrictions exploit the difference in bidding behavior under two competitive environments, it is crucial that at least one bidder participates in both auctions. 19 For instance, when I 1 = 2 and I 2 = 3, at least one of the bidder in the two bidders auction must participate in the auction with three bidders. In the case of asymmetry, because of the complexity of the system of differential equations defining the equilibrium strategies, it is difficult if not impossible to prove that equilibirum strategies are increasing with competition. Nevertheless, because of the independence of 19 More generally, it is important that we observe at least one bidder s type in both auctions. This is useful in practice as a few types are often entertained in empirical studies involving asymmetric bidders. See for instance Campo, Perrigne and Vuong (2003), Athey, Levin and Seira (2004) and Flambard and Perrigne (2006). 17

20 private values, it is reasonable to postulate that equilibrium strategies are increasing in the number of bidders due to the competition effect. Let s ij ( ) denote the equilibrium strategy for bidder i =1,...,I j, when the number of bidders is I j, j =1, 2. The boundary conditions are s i1 (v) =s i 2(v) =v for i =1,...,I 1 and i =1,...,I 2, and s ij (v) =s i j(v) for i, i =1,...,I j,j =1, 2. We assume that bidders 1,...,I 0 participate to both auctions, where 1 I 0 I 1. Let v i (α) and b ij (α) be the α-quantiles of F i ( ) and G ij ( ) =G i ( I j ), respectively. Instead of (4), we now have at the α-quantile v i (α) =b ij (α)+λ 1 (R ij (α)),i=1,...,i j,j =1, 2 (11) for α [0, 1], where R ij (α) = 1/H ij (b ij (α)) takes values in the range R ij = [0, r ij ] with H ij ( ) = k i g kj ( )/G kj ( ). A straightforward extension of Proposition 3 shows that λ 1 ( ) is identified on [0, max i=1,...,i0 r i1 ], while [F 1,,F I0 ] are identified on [v, v]. Because λ 1 ( ) is identified on [0, max i=1,...,i0 r i1 ], which may be a strict subset of [0, r i1 ], where i refers to a remaining bidder, his private value distribution may not be identified everywhere justifying the partial identification result of Proposition 6. Lemma 3 also extends where the compatibility conditions (7) now hold for each of the I 0 bidders. Asymmetry in Preferences We consider structures of the form [U,F] UR I F R with U = {(U 1,...,U I ) UR I Ii=1 U R,I I} UR. I Given I Iand dropping the superscript to simplify, we obtain for i =1,...,I v i = b i + λ 1 i ( ) 1 H i (b i I) ξ i (b i,u i, G,I), (12) where λ i ( ) =U i ( )/U i( ) and H i ( I) = j i g j ( I)/G j ( I). For each I, the boundary conditions s 1 (v) =... = s I (v) =v and s 1 (v) =... = s I (v) give a common support [b(i), b(i)] for the bid distributions across bidders. Let G R be the set of distributions G(,..., ) such that each marginal distribution G i ( ) satisfies Definition 3 with G(b I)/g(b I) replaced by 1/H i (b I) in (v). Proposition 7: Any structure [U,F] UR F I R with asymmetry in preferences satisfying H i ( I) < 0,i=1,...,I,I Iis not identified.20 On the other hand, the structure [U,F] 20 The assumption H i ( I) < 0 corresponds to an increasing markup v iα b iα in α from (12). If all the 18

21 with the exclusion restriction F ( I) =F ( ) is identified. Namely, U i is identified on [0, max v [v,v] v s i (v)] for i =1,...,I, while F ( ) is identified on [v, v]. Because the α-quantiles (b 1α,...,b Iα ) all correspond to the same α-quantile v α, (12) evaluated at an α-quantile for an arbitrary pair (i, j) of bidders gives the compatibility condition b jα + λ 1 j ( ) 1 H j (b jα I) = b iα + λ 1 i ( ) 1. (13) H i (b iα I) The bid distribution G(,..., ) is then rationalized if and only if (i) Lemma 1 is satisfied with G R and ξ i ( ),i =1,...,I as defined above and (ii) the compatibility condition (13) is satisfied for any pair (i, j) and α [0, 1]. 21 The latter condition reduces the set of bid distributions that can be rationalized relative to the symmetric case and may help in identification. Despite this condition, the nonparametric model is still not identified. Since the proof is more involved than in previous cases, we provide a proof of such a result in the appendix. 22 Under exogenous bidders participation, we assume again that I 0 bidders participate to both auctions, where I 0 1 and that equilibrium strategies are increasing in competition. Equation (11) takes a similar form with v(α) and λ 1 i ( ) replacing v i (α) and λ 1 ( ). A similar argument as in Proposition 3 applies for identifying nonparametrically λ 1 i ( ) on [0, r i1 ] for i =1,...,I 0 from which we can identify F ( ) on[v, v]. Since F ( ) is identified everywhere, following a similar argument as in Lu and Perrigne (2006), the λ 1 j ( )s for the remaining bidders are identified from (12). Specifically, because the v(α)s are identified, we can recover the remaining λ 1 j ( ) on[0, r j ]. Again Lemma 3 extends with the compatibility conditions (13) holding for each of the I 0 bidders and λ 1 i ( ) replacing λ 1 ( ). bid distributions G 1,...,G I are log-concave, this assumption is automatically satisfied. Our requirement is weaker as some bid distributions may not be log-concave. Log-concavity is usually verified on data. 21 Though similar in spirit, the compatibility condition (13) applies within each auction, while the compatibility condition (7) applies across auctions. 22 On the other hand, if (say) bidder 1 participates to all auctions and his utility U 1 ( ) is known, the nonparametric model UR I F R becomes identified as (12) for i = 1 allows to identify F ( ). Thus, evaluated at the α-quantile, (12) for i 1 allows to identify λ i ( ) on [0, max α (v α b iα )]. This result is useful when bidders differ by their sizes and large ones, wich participate to all auctions, can be assumed to be risk neutral. 19

22 Asymmetry in Both Preferences and Private Values This third case involves asymmetry in both private value distributions and preferences. Given the above results, the model is not identified in general. We then consider exogenous bidders participation. Thus, (11) takes a similar form with λ 1 i ( ) replacing λ 1 ( ). Despite the complexity of this case, which has not been considered to our knowledge, it can be shown that the structure [λ i,f i ] is nonparametrically identified for the I 0 1 bidders who participate to both auctions. Specifically, we can apply Proposition 3 to each of these participating bidders to identify nonparametrically λ 1 i ( ) and F i ( ) on [0, r i1 ] and [v, v], respectively. On the other hand, we cannot identify the pair [λ 1 i ( ),F i ( )] for the other bidders. Again Lemma 3 extends with the compatibility condition (7) holding for each of the common bidders where λ 1 i ( ) replaces λ 1 ( ). 5.4 Bidding Strategies Nonincreasing in Competition In the previous extensions, we have assumed that equilibrium strategies are increasing in the number of bidders to simplify the exposition. This may not be always the case. For instance, as indicated previously, affiliated private values may lead to equilibirum strategies that are decreasing in competition for some F(,..., ). In this section, we discuss how our results extend when the equilibrium strategies are nonincreasing in competition. As before, let I 1 <I 2. We assume that for a bidder participating to both auctions his equilibrium strategies s 1 ( ) and s 2 ( ) intersect a finite number of times at most. This excludes the case where these strategies are identical on some open interval of private values. The nonparametric identification of the model is then established through the following steps: Step 1: From the knowledge of G 1 ( ) and G 2 ( ), we can identify the positive values 0 α1 <...<αk 1 at which the equilibrium strategies s 1 ( ) and s 2 ( ) intersect, i.e. such that b 1 (αk )=b 2(αk ). Step 2: Let s j ( ) <s j ( ) on(v,v(α1)),j,j =1, 2. By Proposition 3, for any α 0 (0,α1 ), we can identify λ 1 (u 0 )as + t=0 b(α t ), where u 0 = R j (α 0 ). By continuity of λ 1 ( ), we can also identify λ 1 (R j (α1)) which is also equal to λ 1 (R j (α1)) by the compatibility condition (7). Hence, λ 1 ( ) is identified on [0, max α [0,α 1 ] R j (α)] = 20

23 [0, max{max α [0,α 1 ] R j (α), max α [0,α 1 ] R j (α)}]. Step 3: We have s j ( ) <s j ( ) on(v(α1 ),v(α 2 )). For any α 0 (α1,α 2 ), we let u 0 = R j (α 0 ) and by Proposition 3 we construct recursively α t+1 from the equation R j (α t+1 )=R j (α t ) subject to α t+1 <α t. There are two possibilities: (i) If α t+1 [0,α1], we stop this sequence as we switch to values for which s j ( ) s j ( ). We have λ 1 (u 0 ) = λ 1 (R j (α t+1 )) + t s=0 b(α s ). But λ 1 (R j (α t+1 )) = λ 1 (R j (α t+1 ))+ b(α t+1 ), where λ 1 (R j (α t+1 )) is identified from Step 1 and hence equal to + r=0 b(α r ) for some decreasing sequence {α r} with α 0 = α t+1. Thus λ 1 (u 0 )= + r=1 b(α r) + t s=0 b(α s ), which gives λ 1 (u 0 )= + t=0 b(α t ) by letting α r α t+r. Figure 2 illustrates this case, where j =1,j = 2 and t +1=2. (ii) If α t+1 remains in (α1,α 2) for all t, the sequence {α t } will converge to α1. Taking the limit gives λ 1 (u 0 )=λ 1 (R j (α1 ))+ + s=0 b(α s ). But λ 1 (R j (α1 )) = λ 1 (R j (α1)), where the latter is identified from Step 2. Figure 3 illustrates this case with j = 1 and j =2. By continuity of λ 1 ( ), we can also identify λ 1 (R j (α2)) which is also equal to λ 1 (R j (α2 )) by the compatibility condition (7). Hence, at the end of Step 3, λ 1 ( )is identified on [ min{min α [α 1,α 2 ] R j (α), min α [α 1,α 2 ] R j (α)}, max{max α [α 1,α 2 ] R j (α), max α [α 1,α 2 ] R j (α)} ]. By combining Step 2 and Step 3, λ 1 ( ) is identified on [0, max{max α [0,α 2 ] R j (α), max α [0,α 2 ] R j (α)}]. Step 4: For any k 2 and α 0 (αk,α k+1), we repeat Step 3 and its two possibilities. If α t+1 [0,αk ], we stop the sequence and we have λ 1 (u 0 )=λ 1 (R j (α t+1 ))+ ts=0 b(α s ), where λ 1 (R j (α t+1 )) is identified from previous steps. Thus λ 1 (u 0 )= + t=0 b αt as in Step 3-(i). If α t+1 remains in (αk,α k+1 ), Step 3-(ii) applies and λ 1 (u 0 ) is identified. As before, by continuity, λ 1 [R j (αk+1)] = λ 1 [R j (αk+1)] is identified. Applying a similar argument, the combination of the various steps allows us to identify λ 1 ( ) on[0, max{max α [0,α k+1 ] R j (α), max α [0,α k+1 ] R j (α)}] and hence on [0, max{max α [0,1] R j (α), max α [0,1] R j (α)}] when k +1=K. 21